Answer
The average distance is $\frac{a^2}{2}$.
Work Step by Step
In polar coordinates, we have $D=\{(r,\theta)|0\leq r\leq a,0\leq \theta \leq 2\pi\}$.
The distance of any point $(x,y)$ in $D$ to the origin is defined by a function $d(x,y)=x^2+y^2$.
Find the area of $D$:
$A=\iint_D dA=\int_0^a\int_0^{2\pi}rd\theta dr=\int_0^ar\theta ]_0^{2\pi}dr=\int_0^a2\pi r dr=\pi r^2]_0^a=a^2\pi$
Find the integral $\iint_D d(x,y)dA$:
$\iint_D d(x,y) dA=\int_0^a\int_0^{2\pi} r^2\cdot rd\theta dr$
$=\int_0^a\int_0^{2\pi} r^3 d\theta dr$
$=\int_0^ar^3\theta_0^{2\pi}dr$
$=\int_0^a2\pi r^3 dr$
$=\frac{\pi r^4}{2}]_0^a$
$=\frac{a^4\pi }{2}$
Find the average distance:
$\bar{d}=\frac{\iint_D d(x,y)dA}{A}=\frac{\frac{a^4\pi}{2}}{a^2\pi}=\frac{a^2}{2}$
